Properties

Label 184.19
Modulus $184$
Conductor $184$
Order $22$
Real no
Primitive yes
Minimal yes
Parity even

Related objects

Learn more

Show commands for: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(184, base_ring=CyclotomicField(22))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([11,11,15]))
 
pari: [g,chi] = znchar(Mod(19,184))
 

Basic properties

Modulus: \(184\)
Conductor: \(184\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(22\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: yes
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 184.j

\(\chi_{184}(11,\cdot)\) \(\chi_{184}(19,\cdot)\) \(\chi_{184}(43,\cdot)\) \(\chi_{184}(51,\cdot)\) \(\chi_{184}(67,\cdot)\) \(\chi_{184}(83,\cdot)\) \(\chi_{184}(99,\cdot)\) \(\chi_{184}(107,\cdot)\) \(\chi_{184}(155,\cdot)\) \(\chi_{184}(171,\cdot)\)

sage: chi.galois_orbit()
 
pari: order = charorder(g,chi)
 
pari: [ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
 

Related number fields

Field of values: \(\Q(\zeta_{11})\)
Fixed field: 22.22.339058325839400057321133061640411938816.1

Values on generators

\((47,93,97)\) → \((-1,-1,e\left(\frac{15}{22}\right))\)

Values

\(-1\)\(1\)\(3\)\(5\)\(7\)\(9\)\(11\)\(13\)\(15\)\(17\)\(19\)\(21\)
\(1\)\(1\)\(e\left(\frac{10}{11}\right)\)\(e\left(\frac{2}{11}\right)\)\(e\left(\frac{5}{11}\right)\)\(e\left(\frac{9}{11}\right)\)\(e\left(\frac{3}{22}\right)\)\(e\left(\frac{1}{22}\right)\)\(e\left(\frac{1}{11}\right)\)\(e\left(\frac{17}{22}\right)\)\(e\left(\frac{5}{22}\right)\)\(e\left(\frac{4}{11}\right)\)
value at e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 184 }(19,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{184}(19,\cdot)) = \sum_{r\in \Z/184\Z} \chi_{184}(19,r) e\left(\frac{r}{92}\right) = 0.0 \)

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 184 }(19,·),\chi_{ 184 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{184}(19,\cdot),\chi_{184}(1,\cdot)) = \sum_{r\in \Z/184\Z} \chi_{184}(19,r) \chi_{184}(1,1-r) = 0 \)

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 184 }(19,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{184}(19,·)) = \sum_{r \in \Z/184\Z} \chi_{184}(19,r) e\left(\frac{1 r + 2 r^{-1}}{184}\right) = -4.6774355495+-10.2421551914i \)